Archive for the ‘Mathematics’ Category

Everyone knows that I love tearing down terrible “formula for” stories, but hopefully this one of my own won’t receive the same treatment as it is actually based on some solid maths!

Marathon runners need never “hit the wall” again thanks to a mathematical model that will help them reach the finish line in their best time.

More than 40 per cent of marathon runners will hit the wall during a race, experiencing sudden pain and fatigue as their carbohydrate reserves run low and their body switches from burning carbohydrate to burning fat. So Benjamin Rapoport at the Massachusetts Institute of Technology has given runners an online calculator that will tell them how much carbohydrate they need to consume to have enough for a whole race.

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My latest article for New Scientist is about a mathematical proof showing that it’s always possible to solve a Rubik’s cube in 20 moves or less. Don’t expect to do it by hand though – cracking this puzzle required a supercomputer or two:

It has taken 15 years to get to this point, but it is now clear that every possible scrambled arrangement of the Rubik’s cube can be solved in a maximum of 20 moves – and you don’t even have to take the stickers off.

That’s according to a team who combined the computing might of Google with some clever mathematical insights to check all 43 quintillion possible jumbled positions the cube can take. Their feat solves the biggest remaining puzzle presented by the Rubik’s cube.

“The primary breakthrough was figuring out a way to solve so many positions, all at once, at such a fast rate,” says Tomas Rokicki, a programmer from Palo Alto, California, who has spent 15 years searching for the minimum number of moves guaranteed to solve any configuration of the Rubik’s cube.

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Ever suffered from a limp wrist, or been on the receiving end of a painfully iron grip? Car manufacturer Chevrolet know all about the importance of a good handshake, which is why they’ve developed a complex mathematical equation for their new staff training guide, as that well known science journal the Daily Mail reports.

It’s been a while since I covered this kind of dodgy maths, so let’s go over the basics. “Formula for” stories are seen by PR agencies as a great way to get free press coverage for whatever product they are shilling because the equations can be dressed up as real research. Attaching a “Dr” or “Prof” to your news story is a great way to gain legitimacy, and the media lap it up as another example of what those crazy boffins are up to.

While this is all great for the PR agencies and their clients, it’s terrible for science. These formulas tend to be based on extremely dodgy assumptions and contain variables which can’t be objectively measured. What’s worse, even a simple mathematical analysis usually reveals problems such as division by zero, which can lead to things like cold and lumpy but infinitely perfect pancakes.

With these problems in mind, let’s take a look at the formula for the perfect handshake. It was created by Geoff Beattie, head of Psychological Sciences at the University of Manchester, and is detailed in this the press release:

I’ve broken it over two lines because the thing is so long, and I think that square root is meant to cover the entire equation, not just the first term, but the press release isn’t very clear. We’ve also got a definition for the many variables, along with what I assume is their optimal values:

Both the formula and its variables are looking really dodgy. I’ve literally no idea what terms like {(4<c>2)(4<du>2)}² are meant to mean. I can only think that the angular brackets denote some kind of average, but then why do they only apply to some of the variables? Are those 2s actually meant to be ²? In which case you can rewrite the whole term as (2<c><du>)4, which is at least a little bit simpler.

I also take issue with using two letters to stand in for one variable, because they can be confused for two separate variables multiplied together. Measuring “verbal greeting” and “vigour” doesn’t mean that both of your variables have to start with a v – real mathematical equations make extensive use of Greek letters in an effort to solve this exact problem. But even if this equation was beautifully formatted, it would still be rubbish.

All the measurements are completely subjective, and the scales of 1 to 5 indicate the data behind the equation was probably collected from a survey. This even includes variables such as temperature, which can easily be measured scientifically. Remember, subjective measurements are one of the hallmarks of a “formula for”.

I emailed Beattie yesterday to ask how the formula was created, but as he is yet to reply I can only speculate. I think what he has done is ask people a bunch of questions about handshakes, and then tried to fit their answers to some kind of least-squares model, as indicated by the squares and square root in the formula. This method gives you a great equation for “explaining” the data you’ve gathered, but doesn’t necessarily tell you anything about the phenomena you’re examining.

If that is the case, I still don’t understand how the formula is meant to work. You’d expect that the perfect handshake would have a maximum value of PH, and since there is no division or subtraction involve, that just means slotting in the maximum values for all your variables. The optimal values in the press release include a few 3s and 4s though, so PH isn’t going to be maximum. Hmm.

As with all “formula for” stories the maths behind the perfect handshake formula just doesn’t add up, yet it’s being interpreted as a serious piece of research. Comments on the Mail story such as these two show just how much damage this can do to people’s impressions of science:

“So, most of the country is out of work desperately trying to survive and these idiots are getting paid, what – to study handshakes? Sack these people immediately!”

How much time did the nutty professor spend on this useless bit of information?

Mathematical models and equations are a fantastical tool for understanding the natural world around us, but they have to be based on sound assumptions and decent science – things that “formula for” stories such as this almost invariably lack.

In the past I’ve linked to all kinds of periodic tables, from the edible to the audiovisual. Now, someone’s gone all meta and created a periodic table to list all of these periodic tables:

You can see a larger version here, complete with links to all the other tables.

And you think your job is tough…

Popular Science has drawn up a list of the ten worst jobs in science, which includes thankless tasks such as “armpit detective” and “whale slasher”. Don’t let them put you off pursing a career in science however, as the list also reveals the best job: “multispecies baby tickler”. Where do I sign up?

Fire! De der deeeer, der der…

A Ruben’s tube is a nifty demonstration of standing waves with a healthy dose of burnination:

A really geeky maths joke

I probably find this joke far more amusing than I should:

An engineer, a physicist and a mathematician find themselves in an anecdote, indeed an anecdote quite similar to many that you have no doubt already heard.

After some observations and rough calculations the engineer realizes the situation and starts laughing.

A few minutes later the physicist understands too and chuckles to himself happily as he now has enough experimental evidence to publish a paper.

This leaves the mathematician somewhat perplexed, as he had observed right away that he was the subject of an anecdote, and deduced quite rapidly the presence of humour from similar anecdotes, but considers this anecdote to be too trivial a corollary to be significant, let alone funny.

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Here’s an extract from an article I wrote for New Scientist in honour of Pi Day today.

The stars overhead inspired the ancient Greeks, but they probably never used them to calculate pi. Robert Matthews of the University of Aston in Birmingham, UK, combined astronomical data with number theory to do just that.

Matthews used the fact that for any large collection of random numbers, the probability that any two have no common factor is 6/pi2. Numbers have a common factor if they are divisible by the same number, not including 1. For example, 4 and 15 have no common factors, but 12 and 15 have the common factor 3.

Matthews calculated the angular distance between the 100 brightest stars in the sky and turned them into 1 million pairs of random numbers, around 61 per cent of which had no common factors. He got a value for pi of 3.12772, which is about 99.6 per cent correct.

A serious science survey?

The BBC reports that one in 10 children believe the Queen invented the telephone, while others suggest Charles Darwin and Noel Edmonds. The results come from a survey of 1,000 school kids, but rather than despairing at the state of science education, I’m actually amused by this story.

These types of articles seem to crop up fairly often, with children giving nonsensical answers to questions about historical facts. Everyone always interrupts them fairly seriously, but I think it’s far more likely that the kids are just having a laugh.

High-gravity lava lamps

Would a lava lamp work on Jupiter? There’s only one way to find out – build a giant, semi-lethal centrifuge out of Meccano, and take your lamp for a spin:

I wrote this piece for the Guardian as part of their Valentine’s Day coverage:

Steamy love poems are always popular around Valentine’s Day, but can a few lines of tender verse really make people hot under the collar? Researchers at Aberystwyth University attempted to find out earlier this week, using thermal imaging cameras to take the temperature of volunteers reading the work of Romantic poets.

The experiment is a collaboration between the arts and the sciences, led by poet Richard Marggraf Turley from the Department of English and Creative Writing and Reyer Zwiggelaar from Computer Science. They asked six volunteers from each department to silently read 12 love poems, while a slightly less amorous text about thermal imaging served as a control. As the participants pored over poems, including Bright Star by John Keats and To His Coy Mistress by Andrew Marvell (both are reproduced in full below), thermal cameras monitored their faces for any change in temperature that could reveal their true feelings.

A number of newsoutlets have run stories on a formula for finding your “Optimal Proposal Age”, based on a press release from the University of New South Wales. Far from being a new result, it’s actually a repackaging of an old mathematical puzzle known by a variety of names, including the marriage problem.

Imagine you’ve decided to search for the perfect partner by going on 100 blind dates. After each date you decide whether you want to marry the potential suitor, and if you choose not too you can never see them again. Contrived, but then this is a maths puzzle!

How do you pick your partner? If you wait until the end of all 100 dates, you’ll be stuck with whoever is on the end of the list, whether you like them or not, but if just go for the first person you like then you could be missing out on someone who is a better match. It turns out that the best strategy is to see the first 37 potentials, then pick the next one who is better than those 37. Not the most romantic approach, but at least it makes for a quirky Valentine’s Day news story I suppose.

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Finding it difficult to meet your perfect partner? According to the Daily Mail, a”maths genius” can explain with a “baffling” equation. That’s right, it’s the first “formula for” story of 2010!

The Mail and others have leapt on a rather silly paper by Peter Backus, a University of Warwick economist. He’s used the Drake equation, which was originally intended to estimate the number of alien civilizations in our galaxy, to explain why he doesn’t have a girlfriend.

You can visit Wikipedia for an explanation of the Drake equation, or alternatively check out Colin’s dissertation for the full details. The basic idea is to break down all the requirements for alien life in to individual factors, such as the chance of a star having planets or a planet supporting life, then multiply them together to get the number of civilizations out there in space. Trouble is, we don’t have very reliable evidence to back up most of the figures, so estimates vary wildly.

Backus has used the same principal to find his perfect woman, and “discovered” that there are only 26 women in the UK that are suitable for him. That’s a one in 285,000 chance of meeting “the one”, apparently. Of course, the exact same criticism of the Drake equation can be applied here – most of his numbers are entirely subjective and not backed up by evidence. Pick some different numbers, and you’ll come up with an entirely different answer.

I can’t really blame Backus for his formula, as it’s not like he’s trying to sell anything or has got the maths wrong. What I find annoying is the way the media leaps on the figure of “one in 285,000″ as an absolute fact, and describes maths no more complicated than multiplication as if it were some sort of advanced calculus that should only be attempted by a genius. Let’s just hope no one discovers the ancient art of “division”, or our heads just might explode.

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Hello! It’s 2010, and I’m finally back. I had intended an earlier return to blogging here at Just A Theory, but unfortunately a rather serious computer failure held me up. The hard drive in my PC died, causing Windows to become corrupt and refuse to boot. As you can see, I attempted some minor brain surgery in an effort to revive the poor machine:

I actually had some success, and after more than 12 hours of work was rewarded with this rather understated error message:

Quite. Sadly, in the end I had to say goodbye to my faithful old PC and buy a new one, complete with Microsoft’s latest operating system, Windows 7. It’s quite different to the Windows XP I’m used to, especially as I’d disabled most of XP’s bells and whistles to make it run like Windows 2000. Essentially, I’ve been using the same operating system for an entire decade, and now I’ve been forced to change some long-held habits!

All of which leads me on in a fairly rambling way to what I had originally intended to talk about at the start of 2010 – whether we’re now living in a new decade. The media seem pretty convinced that we’ve abandoned the “Noughties” in favour of the “Teens”, but the maths says otherwise – it won’t be until the end of 2010 and the start of 2011 that we enter the next decade.

It’s the same argument that you probably tired of in the years leading up to December 31st, 1999. At the time, mathematicians said that millennial celebrations should be put off until the start of 2001, while the rest of the world largely ignored them.

Simply put, our calendar system starts at the year 1 AD, not the year 0 AD. One year later is 2 AD, ten years later is 11 AD, and two-thousand years later is 2001 AD. So, new decades start with years ending in a “1″.

But when we speak of the Noughties, we obviously mean the years 2000 to 2009. The year 2010 can’t be a Noughtie, because it doesn’t have a 0 in the right place. And hang on a moment, isn’t the calendar based off the life of Jesus, a man whose date of birth we know very little about? And lets not even start on the missing 11 days of September 1752.

Given the human desire for patterns and our fondness of round numbers, it’s probably best if we stick to celebrating 2010 as the new decade – it’s no less arbitrary than any other choice. Even so, I can’t help wanting to go with 2011. It may be ugly, but it’s mathematically correct!

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My latest article for New Scientist is about a new method for mapping the globe:

A new technique for unpeeling the Earth’s skin and displaying it on a flat surface provides a fresh perspective on geography, making it possible to create maps that string out the continents for easy comparison, or lump together the world’s oceans into one huge mass of water surrounded by coastlines.

“Myriahedral projection” was developed by Jack van Wijk, a computer scientist at the Eindhoven University of Technology in the Netherlands.

“The basic idea is surprisingly simple,” says van Wijk. His algorithms divide the globe’s surface into small polygons that are unfolded into a flat map, just as a cube can be unfolded into six squares.

Cartographers have tried this trick before; van Wijk’s innovation is to up the number of polygons from just a few to thousands. He has coined the word “myriahedral” to describe it, a combination of “myriad” with “polyhedron”, the name for polygonal 3D shapes.

You’ll find the rest at New Scientist, along with some nifty images and video.

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Cliff Arnall is back, and he’s got enough dodgy “formula for” stories to see us through til Christmas. The man behind the worst and best days ever has now come up with a formula for the “perfect toy”. As with all the finest scientific research, you can find the details in the Daily Mail.

If you don’t remember him, Cliff Arnall often pops up in to the media peddling mathematical nonsense. The Mail bill him as “Professor” Arnall, which is a new one, but it’s not entirely clear which institution he’s from. Certainly not Cardiff University, who have made repeated attempts to distance themselves from Arnall after he left their employ as a part-time tutor.

Let’s have some fun playing with the formula then. The Daily Mail have a handy explanation:

All the variables in the left column are basically arbitrary scores out of 5, and thus fairly meaningless. In the right column, T, L and C are at least all quantifiable, in that we can assign a meaningful value to them. Multiplying T by L is actually fine, because both of these variables use units of time. The problems start when you divide by the square root of C.

Quick anyone, what’s the square root of £1? I might as well ask for banana divided by orange – neither question makes mathematical sense, because there is no such thing as the square root of currency.

Our old friends zero and infinity make an appearance as well. If a toy is free, it doesn’t matter if you give it 0 out of 5 for everything else, because as long as your child plays with it for even a second, it’s going to have infinite play value. Dividing by smaller and smaller values of C makes the last term in the equation grow rapidly, completing dwarfing the others. In other words, Cliff Arnall’s perfect toy is crap and worthless. Just like his formulas then.

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This strange object is the Mandelbulb, a 3D version of the famous Mandelbrot set fractal. I’ve written an article about it over at New Scientist, so go check it out, along with a gallery of more 2D and 3D fractals.

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With banks being bailed out all over the place these days, many people are asking themselves why those in charge get paid such high salaries. Are CEOs really worth their million pound bonuses? Not according to Venkat Venkatasubramanian, who has calculated that US chief executives get paid nearly 130 times what they should.

As a professor of chemical engineering at Purdue University, Indiana, Venkatasubramanian seems an unlikely candidate to dictate CEO salaries. It turns out that the maths behind thermodynamics, the study of heat and energy, can also be applied to economics.

The trick is to redefine a concept called entropy. In thermodynamics entropy measures the disorder of a system. Imagine a box full of gas particles. If all the particles are packed into one corner, the system has low entropy. If they are spread out and zooming all over the place, it has high entropy. The laws of thermodynamics mean that entropy always increases over time.

What does that mean for CEO salaries? Venkatasubramanian realised that entropy could be seen as a measure of “fairness” in economics. According to the laws of supply and demand, as markets evolve salaries and the price of goods should move towards the most fair situation for everybody.

With this economic equivalent to entropy, Venkatasubramanian found that salaries should follow a lognormal distribution, a particular way of measuring the spread of data. When he compared his theory to data from income tax returns, he found that the model fit closely for the bottom 90-95% of salaries. In other words, the 5-10% at the top are getting more than their fair share.

According to the lognormal distribution, CEOs should be paid a little over 8 times more than the average employee. Looking at the salaries of 35 CEOs from top Fortune 500 companies, Venkatasubramanian discovered that their pay was on average 1,057 times what a regular employee earns – around 129 times the ideal value.

Of course, not all CEOs pull in such vast amounts. Interestingly, investment guru Warren Buffet takes a salary of $200,000, which is about 8 times what the average employee of his company Berkshire Hathaway earns.

Venkatasubramanian points out that his figures only work for large corporations – the heads of smaller entrepreneurial start-ups will clearly be worth more than the few staff they employ. Still, he hopes that his research will be useful to governments and regulators in assessing CEO salaries, and ensuring a fair deal for all.

For some reason the Independent have decided to publish the mother of all “formula for” stories – ten examples of the best worse science reporting there is. They include ones I’ve written about before, like the formula for the perfect pancake,but also a bunch I’d not previously seen. The best has to be the equation for the perfect sandcastle, which is OW = 0.125 x S. In other words, one part water, eight parts sand.

Lunch time at the Periodic Table

This photo of a literal Periodic Table has been doing the internet rounds recently:

Turns out it’s a piece of art work at Wake Forest University in North Carolina. It was created by two student in 2003, Nazila Alimohammadi and Anna Clark. Nice work – I’m always up for a good pun!

From coffee to carbon

Also floating about this internet this week was this interactive illustration of the size and scale of various cells from the University of Utah. Starting from a coffee bean and a grain of rice, you can zoom past human cells, bacteria and viruses before ending up at a single carbon atom. Zooming out is just as fun!

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Apologies for my lack of posting this week, I’m once again hepped up on Lemsip as I battle against a cold. My fellow bloggers have done a great job at picking up the slack, but I still have a collection of interesting links from the past week. Here we go:

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As anyone who reads Just A Theory should know by now, “formula for” stories are usually nothing more than thinly veiled PR that newspapers happily print for free, but they don’t get much worse than this:

Say hello to Phillippa Toon, proudly displaying her formula for the perfect night out. Phillippa is a biology student at Leeds University, and also holds the estimated position of “VKendologist”.

For those not up on their alco-pops, VK is an unpleasant mix of vodka, sugar and E numbers served in pubs and clubs across the country. The drink is owned by Global Brands, who it seems placed adverts on Facebook in the hopes of attracting Britain’s brightest minds to figure out the “formula for fun”. The advert read:

“Wanted! Talented maths or science student or graduate to spend the summer literally discovering the formula of fun. Must be over 18 years of age like bars, clubs and pubs and be prepared to have a fantastic time in the quest for knowledge, science and the pursuit of the perfect night out.”

According to the Yorkshire Evening Post, Phillippa was one of “hundreds of mathematicians and fellow scientists” vying for the chance to make up a such a formula.

For once, I can’t fault the maths too much. Yes, the measurements are completely subjective (check out the calculator on the VK website for full details of the variables in the formula), but at least there’s no division by zero leading to the likes of infinitly bad pancakes.

What really gets me about this is how shameless it is. Get a pretty girl, dress her up as a scientist, and gush about the “experiment” she conducted using “maths and science”. She’s not even the usual “expert scientist” they wheel in for these things, she’s still a student. Why do we let companies get away with this? Why do newspapers insist on printing these stories full of nothing but cargo cult science?

I know the answer, of course. Newspapers need to fill their pages with content, and a quirky science story that you can lift straight from a press release fits quite nicely. Never mind that it’s based on complete nonsense – since when do silly things like “evidence” or “facts” matter?

By now, my message has become a mantra I am doomed to repeat forever. Do not believe a word of these “formula for” stories; they are adverts, not science.

High-speed running will sap the energy of even the top athletes, but it seems scientists never tire of it. Dutch statisticians have declared the 100-metre sprint could potentially be run in just 9.51 seconds. The current record, set by Usain Bolt in 2008, stands at 9.69 seconds.

If this sounds familiar, its because I wrote not one but two blogposts last year on the very same subject. This time, the researchers used a branch of statistics called extreme-value theory to analyse previous records.

As the name suggests, extreme-value theory is used to answer questions about extreme events. It’s normally used by insurers to calculate the risks of natural disasters, but it seems that a record-breaking sprint can also be classed as “extreme”.

Machines are better than you

Japanese engineers have built a robot that can move faster than the human eye can see. Watch, with the aid of slow mo, how the robotic hand deftly controls balls and sticks as no human can:

LHC will run on half power

Ah, the Large Hadron Collider. It’s been good to Just A Theory, providing a wealth of blogging material from raps to rants, but has faired less well in actually working. Even the classic technological fix, “have you switched it off and on again?” hasn’t worked, because when the LHC boots up again this November, it will only operate at 3.5 TeV, half normal operating power.

The massive ring had to be shut down in September last year after damage caused by an incident that caused the temperature to rise rapidly. The LHD will run through Christmas to let researchers gain experience in running it, and then the power will be boosted to 5 TeV. If all goes to plan, the machine will be shut down again at the end of 2010 to prepare for full power operations of 7 TeV.

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I guess it’s fitting that I should write a story about bacteria whilst feeling ill:

Computers are evolving – literally. While the tech world argues netbooks vs notebooks, synthetic biologists are leaving traditional computers behind altogether. A team of US scientists have engineered bacteria that can solve complex mathematical problems faster than anything made from silicon.

The research, published today in the Journal of Biological Engineering, proves that bacteria can be used to solve a puzzle known as the Hamiltonian Path Problem. Imagine you want to tour the 10 biggest cities in the UK, starting in London (number 1) and finishing in Bristol (number 10). The solution to the Hamiltonian Path Problem is the the shortest possible route you can take.

The moving walkways used in airports actually slow you down, according to scientists in America. Research has found that people reduce their speed when stepping on to a travelator, making the human conveyor belts only marginally faster than walking. This is only true on an empty walkway however, as any congestion will drop your speed to less than a normal walking pace.

Manoj Srinivasan of Princeton University created a mathematical model to investigate the problem. Publishing in the journal Chaos, he found that the conflict between what your eyes see and your legs feel is responsible for the reduction in speed.

Visual cues tell the brain you are travelling faster than your legs are walking, so in order to conserve energy you slow down. This means that using an empty travelator will only save you about 11 seconds for every 100-metre stretch, compared to walking on regular ground.

But as any regular fliers know, airport travelators are rarely empty. Another study by Seth Young of Ohio State University found that delays due to other travellers getting in the way occur so often that you are better off avoiding the walkway all together. “Moving walkways are the only form of transportation that actually slow people down,” said Young, speaking to New Scientist.

Wii-ly good for you

Active video games like Wii Sports can be a good alternative to moderate exercise for children, according to a study published in the journal Pediatrics.

While not a replacement for more intensive sporting activities, scientists at the University of Oklahoma found they were comparably to a moderate walk. Children aged 10-13 were monitored as they watched television, played the Wii and walked on a treadmill. Both gaming and walking increased the number of calories burned by two to three times. As such, the researchers suggest encouraging kids to play active games instead of more passive ones.

Rather than just publishing a paper, myExperiment lets users share data, files, and other information required to understand and reuse research. The site also allows the usual social networking interactions, such as messaging and groups.

Those of you expecting The Tiger That Isn’t to be a book on the evolution of the big cat family, prepare to be disappointed. The book’s subtitle, “Seeing through a world of numbers”, gives the game away – it’s about maths. More specifically, The Tiger That Isn’t exposes the common misuse and abuse of numbers by politicians, government institutions and the media.

Don’t be too downhearted though, because Blastland and Dilnot, the creator and former presenter of Radio 4′s excellent More Or Less programme on statistics, have written a fantastically interesting book based on their knowledge from the show.

The unusual title refers to the human capacity for pattern recognition. We have evolved the powerful ability to identify patterns, and to notice deviations from those patterns. This important skill allowed our ancestors to see, for example, the distinctive stripes of a tiger in the jungle and run away to safety.

Pattern recognition comes at a cost however. Sometimes our over-active brains will see the tiger that isn’t – a chance occurrence of light shining through the long grass that gives the impression of a non-existence tiger.

The Tiger That Isn’t guides readers through common mistakes in the use of statistics with examples plucked from the headlines. An NHS deficit of £1bn sounds immense, but it works out as less than 1% of the total NHS budget, and just £16 per head. League tables are revealed as effectively useless, with schools shooting up and down based on little more than random chance. And as we already know, the media is notoriously bad at reporting health risks.

If you’ve ever enjoyed an episode of More or Less, read a newspaper and wondered where all the numbers come from, or even just uttered the phrase “lies, damned lies and statistics,” this is a book you will enjoy. In addition to being entertained, you’ll finish The Tiger That Isn’t with a much better understanding of what numbers can and can’t tell you. Read it.

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Cliff Arnall is the king of the “formula for” story. Earlier this year I wrote about his equation for calculating the date of Blue Monday, his self-styled worst day of the year.

At the time I failed to mention that Arnall actually trots out this rubbish not just once, but twice annually. When summer rolls round, it’s time for the happiest day of the year, which according to Arnall’s formula was yesterday.

The “story” was picked up by the Telegraph, Daily Mail, and Sun. Fact-checking obviously doesn’t occur on the happiest day of the year, because it seems that Arnall is still dining out on Cardiff University’s reputation, despite the institution making it very clear he only worked there as a part-time tutor.

I suppose its time to take a look at the formula now, but by this point do you really need me to tell you it’s nonsense? Here, in all its glory, is the “complicated equation” needed to calculate a day’s happiness rating, along with the variable definitions:

O + (N x S) + Cpm/T + He

O: Outdoors

N: Nature

S: Social interaction

Cpm: Childhood memories of summers

T: Temperature

He: Holidays

Not sure about the difference between outdoors and nature, and surely the value will be the same for each day; O = N = 1, unless there is a second outdoors that I don’t know about. Social interaction could actually be quantifiable, perhaps the number of conversations in a day, but it’s pretty unclear.

Cpm and He are both very bad notation. What is wrong with just C and H? The extra letters don’t add anything, they aren’t even an abbreviation, but they could easily be confused for additional variables. I guess this way looks more “scientific”.

In fact, the only scientifically measurable variable, temperature, is what makes this “formula” fall apart. Assuming you have at least some memory of your childhood, Cpm/T will rapidly grow to infinity as the temperature drops to 0 °C and completely dominate anything else in the equation.

I don’t know about you, but I thought it was pretty warm out yesterday. It seems that Arnall’s Blue Monday, January 19th, would be a much better candidate for happiest day of the year according to this formula. Maybe he accidentally got his bullshit mixed up with his bollocks, and gave us all the wrong date. Now that’s a thought that makes me smile.

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Gangs of teenagers roaming the land are generally bad news, but not when it comes to ravens. Juvenile ravens hunting in packs have gotten some scientists very excited, as this behaviour was predicted by a mathematical model before ever being seen in the wild.

Dr Sasha Dall lectures in mathematical ecology at the University of Exeter, and in 2002 set out to solve an evolutionary puzzle: why do young ravens share their food? Natural selection tells us organisms should only help themselves and their relatives. It seems that no one told the ravens.

Typically, juvenile ravens spend their winters drifting in and out of communal roosts. They scavenge for food, usually sheep carcasses, by themselves. Having found a tasty meal they return to the roost and recruit other ravens for a feast the next day. These shared dwellings can house up to 100 individuals, but they don’t stick around. Each bird will move on every few days to another roost and probably won’t encounter their former roommates again.

“From an evolutionary perspective, this is a bit weird,” says Dall. The ravens are unrelated so will not pass on their genes by helping out others. They also don’t encounter the same individuals often enough to build up a sense of co-operation. Using a technique called game theory, in which many different strategies are played out, Dall built a model to explain this unusual behaviour.

The favoured hypothesis amongst ecologists was roosts act as a kind of “information centre” to the advantage of all the juveniles. Individual birds are unlikely to find a carcass by themselves, but if every bird shares information about food locations then they all benefit.

Dall’s model showed that this strategy emerged naturally when ravens try to maximise their access to food. “In the long run, they find more carcasses than they otherwise would,” he says. Bringing a few friends along also allows young birds to chase off any adults who might lay claim to carcasses in their territory.

Problem solved then – except the model didn’t provide just one answer. “I did manage to predict this typical behaviour, but my model came up with another evolutionarily stable strategy,” explains Dall. According to the model, gangs of juvenile ravens should also fly around looking for food together, and never roost in the same place twice. But no-one had ever seen this kind of behaviour.

Perhaps this would have been dismissed as purely mathematical curiosity, if weren’t for Jonathan Wright, professor of biology at the Norwegian University of Science and Technology. Wright was studying a large raven roost in North Wales when he noticed that the juvenile birds were organising themselves into hunting packs, just as Dall predicted.

“I was surprised to discover that this behaviour had been observed somewhere,” says Dall. The variables used in the model, such as the size of the ravens’ search area, matched the real world exactly. The two scientists wrote up their findings in a joint paper, published earlier this year in the journal PLoS One.

So what will Dall turn his mathematical predictions to next? “The evolution of animal personality differences,” he says. Dall plans to investigate why animals of the same species behave differently within social groups. Perhaps game theory has the answer.

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A policewoman has come up with a formula designed to increase public confidence in the police. I would have gone with catching criminals and not accidentally killing members of the public, but then what do I know about policing?

Chief Constable Julia Hodson of the Nottinghamshire Police suggests that her formula CE+CI+CS+VCxC = PC is the solution to policing problems. A quick run down of the variables:

CE: Community Engagement

CI: Critical Incidents

CS: Customer Satisfaction

VC: Volume Crime

C:Communication

PC: Public Confidence

You know the drill. Like all “formula fors” we have unquantifiable variables, nonsense algebra, and a completely useless equation. Hilariously, the Daily Mail describe the formula as an “Einstein-style mathematical equation”. Maybe it’s all the “C”s? Who knows.

If you could somehow measure all of these variables, the formula still doesn’t make sense. Why do you multiply Volume Crime by Communication? What on earth is that meant to mean? Hodson has degrees in both law and social policy, but along with everyone else offering “formula fors”, she could probably do with retaking GCSE Maths.

My new friends, the TaxPayer’s Alliance, have also criticised the formula. They make a bit more sense than when they were quacking on about ducks, with TPA Research Director Matthew Sinclair offering this:

“With the high crime rates in Nottinghamshire the Chief Constable’s time might be better spent working out how to bring criminals to justice rather than concocting dodgy algebra that wouldn’t pass muster even in a grade-inflated GCSE exam.

“This is exactly the kind of nonsense that makes the public wonder whether the police share their priorities, and undermines the public confidence which the formula is supposed to bolster.”

The TPA seem to be worming their way in to a number of news stories at the moment. An organisation to watch out for I think.

As for Chief Constable Julia Hodson and her nonsense formula, it appears that Nottinghamshire police are currently looking for a Scientific Support Manager Opportunity. They want someone to “drive the strategic direction of scientific support and deliver continuous improvements in the quality of forensic service provided to colleagues and the people of Nottinghamshire.” Perhaps providing a few maths lessons on the side wouldn’t hurt either.

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I was going to write about the AQA exam paper today, but decided to push ahead with it yesterday because people were asking to see the paper. As such I’ve only got a little “bonus” link on offer for you this morning: the MegaPenny Project.

People often have difficult wrestling with big numbers like a million or a billion. It’s very hard to picture a billion of anything, but billions and even trillions are tossed around the news all the time these days. I found the MegaPenny Project a number of years ago, and it’s really great for visualing these types of large numbers. Starting with a single US penny the site builds to a massive quntillion, 1 followed by 18 zeros, in pennies by using handy comparisons like a person, a school bus, or the Empire State Building. A bit more than a small chunk of change then.

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One of my friends is a maths teacher, and we’ve often discussed the many problems with maths education in this country. I’d never cut it as a teacher, but it’s still clear to me there is something wrong with the numeracy levels of the general UK population. New research shows that this could be because we’re teaching maths backwards.

The standard way of teaching maths starts out with a few examples before moving on to generalisations. In other words, you learn that 2 x 3 = 6 before moving on to a x b = ab. A study by psychologists Bethany Rittle-Johnson and Percival Mathews of Vanderbilt University in Tennessee has that this may not be the best way for children to learn.

“Teaching children the basic concept behind math problems was more useful than teaching children a procedure for solving the problems – these children gave better explanations and learned more,” Rittle-Johnson said.

“This adds to a growing body of research illustrating the importance of teaching children concepts as well as having them practice solving problems.”

The study, published in the Journal of Experimental Child Psychology (which may or may not be this one as it seems to be six months old…) showed that children who were just taught how to solve problems without the concepts behind them found it difficult to adapt to new problems. Those who understood the concepts however were able to figure out the problems for themselves.

Will this research lead to a change in maths education? I hope so. Mathematician and Professor for the Public Understanding of Science Marcus Du Sautoy has likened current methods to teaching kids scales and arpeggios without actually letting them play music. A more conceptual view of mathematics would be a welcome move away from this.

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How many people do you need to have in a room before it’s more likely than not that at least two of them share the same birthday? As today is my 23rd birthday it’s particularly suitable for a post on this interesting mathematical puzzle – because the answer just happens to be 23.

“Surely not?” is most people’s response, because 23 just seems too low. As there are 365 days in the year, common sense would suggest that you’d need a much higher number of people to give a 50% probability of a shared birthday. The birthday “paradox” isn’t really a paradox, but rather a great illustration of how common sense can let us down.

It works like this. If I’m in a room with 22 other people, that means there are 22 chances that one of them shares my birthday and is also celebrating today. Here’s the catch: the same thing goes for everyone else in the room. That means there are 253 chances in total, because we have (23 x 22)/2 = 253 pairs in the room. The division by two is to avoid counting each pair twice, if you were wondering.

What are the odds that I have a different birthday to just one person? In other words, if I meet someone at random in the street, how likely is it they won’t have been born on 6th April, but some other day instead. If we ignore leap years, and assume that all birthdays are equally likely, there are 364 other days they could have been born on. That means there is a 364/365 chance they don’t share my birthday, which works out around 99.7%. Sounds about right – after all, it’s pretty likely we have different birthdays.

A handy trick often used in these type of calculations is to work out the probability that the opposite of what ever you are interested in happens, and use that to work out the probability that it does.

In this case, we can work out the likelihood that no-one shares a birthday. We already know this figure for one pair, it’s 364/365 as discussed above. To calculate the probability for 253 pairs, we simply multiply this number by itself 253 times.

Reaching for a calculator, we find that (354/365)253 = 0.4995, roughly. That’s the probability that no-one shares a birthday. To find the probability that at least two people do (it could be more) we just subtract this from 1 to get 0.5005, or just over a 50% chance.

You might be wondering how many people you need in a room for a 100% chance of two shared birthdays. That’s more intuitive – with a massive room containing 366 people, you’re guaranteed a match because there are only 365 birthdays! We can however get a 99% chance of a match with only 57 people, using the same method I’ve just described.

With the rise of social networks like Facebook, we can conduct experiments into the birthday paradox quite easily. If you’re logged in, this link should take you to a list of your friend’s birthdays. I’ve got 113 Facebook friends, which means there is a 99.9999996% chance at least two of them share a birthday. Indeed, there are nine shared birthdays, including a four-way share one month ago on 6th March! Not bad. I’m off to eat some cake to celebrate.

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I’d make a pretty rubbish goalie, as I almost let this “formula for” story slip right past me. Last week, the Sun reported that “boffins” have found the formula for a penalty kick that will score every time.

According to “university eggheads” the ball must be kicked at over 65 mph, with a run-up of five or six steps at an angle of 20 to 30 degrees on the ball. When it reaches the goal line, it must be exactly half a metre from both the crossbar and the nearest post. The Sun even provide this handy diagram:

The "perfect" penalty.

Now, I’m no football expert, but if the goalie dives to the same side that the ball is aimed at, there is definitely going to be a greater than 0% chance that they occupy that same space half a metre from the goalposts. That puts a hole in the researcher’s “100% success” claim straight away.

Oh, I can’t even pretend any more. This isn’t remotely research – it’s advertising. Despite The Sun’s “exclusive” label, the story appeared much earlier this month on the Sky Sports website. Professor Tim Cable of Liverpool John Moores University found the “formula” – which isn’t actually a formula mind, just a description – using Sky+HD, that well known piece of research equipment.

It gets worse. The perfect penalty formula was actually “discovered” almost three years ago, according to this BBC article. Back then it was mathematician Dr David Lewis, again of Liverpool John Moores University, who made the shocking breakthrough. He actually provided a formula as well: (((X+Y+S)/2)x((T+I+2B)/4))+(V/2)-1.

I’m not even going to bother breaking that mess down. It’s clear that this “formula for” story is making money for someone though, and based on my extensive research of two samples I predict it will next show up around December 2011. Of course if the researchers at Liverpool John Moores need to make a quick buck a little sooner than that, we could see it as early as the Euro 2010 cup. Not that I’m being cynical or anything.

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Yes, today is Pi Day, a day for people all around the world to get together and celebrate the number π.

A pi pie, of course.

As I’ve discussed before π is an irrational number, meaning that when expressed as a decimal it goes on forever. I normally stick to 3.14 as the value for π. This explains why March 14th is designated Pi Day, as in the American date system it is 3/14. For those of us elsewhere August 22nd is perhaps a more suitable Pi Day, since 22/7 is a commonly used approximation for π.

If you think about it, π is a pretty strange concept. Why should the circumference of a circle divided by its diameter be this ethereal number that we can’t even write down? The symbol “π” is the only true way of expressing π, because any decimal representations – even this file of 1.25 million digits of π isn’t really correct – although the level of approximation is far beyond what you would ever need for practical calculations!

When I first found out about π, I wondered why we couldn’t just say it was 3. Surely, I thought, something could be done to tame this beast of a number. Indeed, in 1897 an effort was made in Indiana to legislate π in to submission, decreeing it equal to 3.2!

Such efforts are in vain. It is a truism to say “π = π” of course, but it’s a fact that we can never change. Perhaps in some strange parallel universe π = 3, but it’s hard for me to even wrap my mind around.

I guess then we’re stuck with π. It’s no bad thing really, as π can be an incredibly useful number in all sorts of calculations. It even has a starring role in my favourite, Euler’s equation. So, have a piece of pie on Pi Day, but be sure to measure the ratio of the circumference to the diameter before you tuck in. If it’s not π, something has gone wrong…

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If you’re not already reading xkcd, then you should be. It’s full of gems like this one:

I sometimes wonder whether the first ten minutes of every school science class shouldn’t be just the chanting of “correlation does not imply causation” until everyone has it burned into their brains. So many misunderstandings of science would be cleared up.

Autism on the rise? Increased use of the MMR vaccine? Yes, there is correlation. Without an underlying mechanism however, there is no causation – and that’s where the science happens. We don’t really care that two things are happening at the same time unless there is something connecting the two. If there is, well designed experiments will help us figure it out.

Sometimes events are just correlated, nothing more. So, the next time someone tells you that Facebook gives you cancer or some other nonsense disguised as statistics, ask them to explain how exactly one causes another. Now, excuse me whilst I return to my morning chant…

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It’s Pancake Day, or Shrove Tuesday for the Christians amongst us. Yesterday the Daily Mail published a story that began:

“With Shrove Tuesday tomorrow it was perhaps inevitable that an eager scientist would apply their skills to creating the perfect pancake.”

I’m tempted to re-word this:

“With Shrove Tuesday tomorrow it was perhaps inevitable that the Daily Mail would run some nonsense about a formula for the perfect pancake.”

Yes, we’ve got another one. Today’s formula faker is Dr Ruth Fairclough, who “worked out the food formula because her two daughters loved eating pancakes so much.” It appears that in between culinary calculations, she is part of the Statistical Cybermetrics Research Group at the University of Wolverhampton, who specialise in downloading and analysing large amounts of data from the internet. With that in mind, on to the formula:

100 – [10L - 7F + C(k - C) + T(m - T)]/(S – E)

Ooh, that’s big and scientific looking. Check out all those variables! What do they mean? Here’s the run down:

L: the number of lumps in the batter

C: the consistency of the batter

F: the “flipping score”

k: the ideal consistency

T: the temperature of the pan

m: the ideal temperate of the pan

S: length of time the batter sits before cooking

E: the length of time the cooked pancake sits before being eaten

The closer the result to 100, the better the pancake is

Phew. Everyone still with me? If you don’t mind, I’m going to skip over my usual complaints of meaningless variables (how do you measure “flipping score”?) and incompatible units, because frankly I’m bored of repeating myself.

As it stands, this formula is unusable. The “ideal” figures, k and m, are both constants, which means that they are the same each time – as you would expect, because if something is ideal then it shouldn’t change! An example of a constant in a real scientific formula is the “c” in the famous E = mc2. Here, c stands for the speed of light, which even Google knows is around 300 million metres per second. We’ve no idea what k and m are in the pancake formula however, so there is no way of evaluating it.

Even if you could, the construction of the formula contradicts itself. Take the term C(k-C), which obviously has something to do with consistency (no pun intended). It’s one of many terms in the formula that is being taken away from 100, and since we want our result to be as close to 100 as possible, we should probably try and make C(k-C) as small as we can.

Using a mathematical technique known as differentiation, it is easy to work out there is no minimum value for C(k-C), but there is a maximum – when C = k/2. Not much use there, but if we assume that C has to take positive values (after all, what does negative consistency mean?) then C(k-C) is at a minimum when C = 0 or C = k. In these cases, C(k-C) will be zero.

Hang on a second. That means that the formula is telling us that in order to get a perfect pancake, with should either strive for the ideal but unknown consistency k, or alternatively, the worst consistency possible (i.e., zero). That doesn’t sound too tasty. The maths is identical for the T(m – T) term, so a pan cooled to 0 °C will mean you are well on the way to perfection.

So, we’ve got our batter with its terrible consistency, and have just taken the pan out of the freezer. How long should we let the batter sit before starting? The worst thing we could do is leave the batter out for a fraction of a second longer than it takes us to start eating the pancake.

This is because as S and E become closer together, the S – E term will approach zero, causing the equation to balloon to infinity. Sit your batter for 10 seconds and your pancake for 9.999999999 seconds, and you’ll have a pretty awful snack on your hands. Conversely, let that pancake have 10.000000001 seconds rest before you tuck in, and you’re approaching culinary heaven because the minus sign is reversed.

Like most of its ilk, this formula is full of holes that are clear to any mathematician. Apologies if I’ve gone off on a slightly technical rant, but I really cannot stand these “formula for” stories, but what concerns me is that Fairclough actually teaches maths at Wolverhampton, or at least did in 2007 – she’s listed as the module leader for a variety of statistical units. If I were in her class, I’d be pretty worried.

“As a footballer, you’re trying to get in line with an incoming free kick. Wayne Rooney is subconsciously solving a quadratic equation every time he works out where to stand in the box. That doesn’t mean he can do it on paper and I’m sure he’s probably forgotten how to do it. But the point is that our brains are evolutionarily programmed to be able to do it.”

I’m never entirely convinced by this type of argument, as I don’t think your brain is really solving equations for you go about your life, but it’s always nice to see a bit of maths promotion. Have a read.

What should the government discuss?

The House of Commons Innovation, Universities, Science and Skills Committee is inviting members of the public to suggest topics for discussion at an oral evidence hearing to be held in a few months time. If you’ve got some scientific grievances that need airing in public, now is your chance.

Any topic under the remit of the Department for Innovation, Universities and Skills will be considered, as long as it is not already covered by an existing enquiry. Your idea must also be examinable in under two hours, and appropriately timed for either April or May.

After reading du Sautoy’s interview above, an idea might be to look into what can be done to stop bankers poaching all the top science graduates. Perhaps I’ll get around to writing it up…

Apocalypse meh

The Met Office Hadley Centre, an government organisation involved in climate change research, suggests that “apocalyptic predictions” about global warming are just as bad as claims that the phenomenon does not exist. Dr Vicky Pope is head of climate change advice at the Met Office, and says that scientists and journalists must stop misleading the public.

“Having to rein in extraordinary claims that the latest extreme [event] is all due to climate change is at best hugely frustrating and at worse enormously distracting. Overplaying natural variations in the weather as climate change is just as much a distortion of science as underplaying them to claim that climate change has stopped or is not happening.”

A common misrepresentation is to extrapolate off only a few years data, which could lead to puclic confusion when scientists’ predictions don’t actually occur, says Dr Peter Stott, a climate researcher at the Met Office.

“The reality is that extreme events arise when natural variations in the weather and climate combine with long-term climate change.”

“This message is more difficult to get heard. Scientists and journalists need to find ways to help to make this clear without the wider audience switching off.”

In a paper appearing in the Journal of Theoretical Biology entitled Duration of courtship effort as a costly signal, researchers Robert M. Seymoura and Peter D. Sozoud use a branch of mathematics called game theory to model a “courtship encounter” between a male and a female.

If you saw The Dark Knight last summer then you’ve seen game theory in action. In a re-working of a classic game theory problem, Heath Ledger’s Joker has rigged two ferries with explosives. On one, the good citizens of Gotham. On the other, a prison-load of thugs and criminals. The Joker, maniac that he is, gives the detonators of each ship to the other ship – so that the citizens can choose to blow up the criminals, and vice versa. He informs them that if they killer their counterparts in the next 30 minutes they will be spared, otherwise he’ll just blow up both ships.

Game theory informs us that the best strategy is for one ship to blow up the other – but of course, this means that both ships will be destroyed anyway, just as the Joker planned. Thankfully, dramatic forces (and Batman) intervene before anyone is harmed. If you want to know more about the maths behind it, a decent explanation is here, but my point is that game theory is a genuine branch of mathematics, and not some crackpot PR nonsense.

The game considered in this paper consists of a male and a female (of any species) engaging in courtship. As time goes on, both parties pay a “cost” for participating in the game. In a human context, this might be a man paying for dinner, whilst the woman he is dating suffers a “cost” to her time – i.e., she might be wasting the evening with an unsuitable mate when she could be finding someone more to her liking. The model also takes into account other species however, for example a male bird singing to a female.

The game ends either when one of the two quit playing (give up to try with someone else) or the female accepts the male as a mate. It is also assumed that males are either “good” or “bad” from the female’s perspective, but she isn’t able to tell good from bad directly. It is only when the pair mate that the female receives a positive payoff from a “good” mate, or a negative payoff from a “bad” one.

The research shows that when the game plays out, a “good” male will participate longer than a “bad” male, allowing the female to weed out a suitable mate: the longer they hang around, the more likely it is that the male will be “good”. Now, this is quite far off from the Daily Mail headline, but the point is that in this case the science is sound. The authors admit that their generalised species model probably doesn’t fit well with humans, especially in a society where contraceptive is readily available. What this research provides is a possible explanation for the evolution of lengthy courtships in many species, including humans. It may not be earth-shattering, but it is science.

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According to the Daily Mail, today is “the most depressing day in HISTORY” – so say the “experts”. Forget your credit crunches, terrorist attacks and military invasions, today is Blue Monday, so we should all apparently feel very sorry for ourselves. Well, I’m quite pleased actually because I get another nonsense formula to bash!

You may remember “worst day ever” stories from years past, because in fact this little gem is rolled out annually. It is supposedly derived from a formula thought up by Cliff Arnall, “formerly of Cardiff University”, but is actually a product of PR company Porter Novelli according to good ol’ Ben Goldacre. The company approached Arnall to put his name to the “research” that was put out as part of a promotion for Sky Travel. Arnall, who Cardiff University have made clear was only a former part-time tutor for them, now seems to make a habit of promoting this rubbish at every opportunity.

On to the formula. The official “Beat Blue Monday” website offers the following formula for calculating the worst day of the year:

The model was broken down using six immediately identifiable factors; weather (W), debt (d), time since Christmas (T), time since failing our new year’s resolutions (Q), low motivational levels (M) and the feeling of a need to take action (Na).

These “immediately identifiable factors” are of course nothing of the sort; notice as well that the variable D is undefined. My usual complaints apply: variables that make no sense (how to you turn “weather” in to a number?) and broken equations (if your motivational level is zero, then the result is infinite), but there is also some nasty abuse of notation here. Na is obviously meant to stand for “need action”, but variables represented like this would normally be part of a series, e.g. Na, Nb, Nc, etc. I guess using the notation in this way makes it look more “scientific”.

Another fault is that although the formula supposedly results in a universal “worst day”, the variables seem to be very individual. Surely “time since failing our new year’s resolutions (Q), low motivational levels (M) and the feeling of a need to take action (Na)” all change from person to person? I was going to try and work back from the result to determine just what they are inputting for the equation, but it’s such a mess it isn’t even worth bothering. Today isn’t Blue Monday at all – I’ve just had a rather good laugh.

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What is it about roundup posts that bring out the puns in me? I think that even science doesn’t have an answer for that one. Have some maths-related nuggets:

Maths failures cost the UK £2.4bn a year

Accounting firm KPMG has found that children who are bad at maths end up costing UK taxpayers up to £2.4bn a year. KPMG say that children who struggle with maths at school are more likely to be unemployed, and claim more benefits whilst paying less tax.

The Every Child a Chance trust is asking businesses to raise £6m in an effect to raise child numeracy. John Griffith-Jones, chairman of both the trust and KPMG, says that the charity has developed a nationwide plan, in which businesses will make annual contribution of £12,000 each to local schools for three years.

A spokeswoman for the Department for Children, Schools and Families responded to the report, saying:

“Let’s be clear: the picture in maths is a positive one…we are leading Europe in maths and science at age 14 and we have risen 11 places in international league tables since 2003 to seventh place.

“Of course there is more we can do and catch up and stretch classes will ensure those falling behind get the additional support they need whilst those who excel are kept motivated.”

Should we be teaching the odds? Probably

Professor David Spiegelhalter of the University of Cambridge believes that children need to be taught “risk literacy” – a knowledge of statistics and probability to help them make important decisions in their adult life.

Speaking to the Times, Prof Spiegelhalter said that risk was “as important as learning about DNA, maybe even more important,” because the human mind has a tendency to latch on to “improbable” coincidence that often turn out to be very possible.

I often point this out to people when unlikely occurrences crop up with this example: if an event has a million to one chance of occurring in a given day, and the population of the UK is about 60 million, then 60 “million to one chances” happen up and down the country daily! The maths doesn’t quite work like that, but it’s a close enough approximation.

As such, I completely agree that ideas about risk should be taught in the classroom. We might then avoid stories such as “beer gives you cancer“, from a couple of weeks back.

Maths: an ideal subject for the lazy

This Times interview with Marcus du Sautoy, the new Oxford University professor for the public understanding of science, is worth a read in general, but I noticed a quote that particularly resonated with me:

Maths, according to du Sautoy, is the ideal subject for anybody who is lazy or has a bad memory (because if you forget something, you can work it out from first principles): “If you do maths, you can get away with hardly doing any work at school and winging it in exams.

This is a notion that I suspect many mathematicians secretly harbour: once you have a grasp of the basics, maths is actually not that hard. Being able to work stuff out from first principles has got me out of a number of tricky exam situations. Another old favourite is starting a problem from both the beginning working forward, and the end working backwards, in the hope that the two sets of calculations will meet up in the middle and save you the trouble of figuring out the problem in the first place!

Of course, there are downsides. Now that I’m on a course full of set readings and essays, I’m dreading actually having to learn and remember facts – I’m not sure I know how!

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Ah, iPlayer. What would I do without you? I didn’t manage to catch the BBC4 broadcast of the first episode of Science and Islam last night, but thanks to the wonderful catch-up service I am able to provide you with a full review. Of course, services like the iPlayer would be impossible without the internet, which in turn could never arisen without first inventing the computer. And what makes computer software tick? Algorithms.

An algorithm is basically a set of instructions, broken down in to simple steps. A computer can follow an algorithm to do pretty much anything, which is why we find them so versatile. As presenter Jim Al-Khalili (a physicist born in Bagdad) tells us, algorithms were invented by a Persian man known as Mohammad ebne Mūsā Khwārazmī, or al-Khwārizmī. Even the word algorithm is derived from his name.

It’s not just algorithms that have been given to us by medieval Arab scholars. The words algebra and alkalis both betray their Arabic origin, but so much of science is attributed to the West. The three part series seeks to unearth the unsung heroes of Islamic science.

The rulers of the Islamic empire realised that with knowledge comes power, and as they spread their influence across the globe the sought out scientific texts from many different regions and cultures. These texts were translated into Arabic, the official language of the empire, which just so happened to be a very scientific language. Originally intended to communicate the teachings of the Koran without misinterpretation, its detailed scripts allowed a precise and unambiguous description of many scientific phenomena.

Much of our modern knowledge can be traced back to this extensive library. In one part of the programme, Al-Khalili visits a modern surgeon to get him to perform a cataract operation by following an Arabic text and using replica instruments from the time. Thankfully for the squeamish the operation is carried out on an eye that has long since been separated from its owner, and the surgeon admits that the instructions are based on sound principles. Indeed, Islamic science provides us with one of the very first anatomical diagrams, showing how the eye is controlled by surrounding muscles.

It’s easy to draw parallels between this programme and an earlier BBC4 one, namely Marcus du Sautoy’s The Story of Maths. Both adopt a sort of travelogue approach, but whilst the earlier programme consisted of nothing but all du Sautoy, all the time, Science and Islam is nicely broken up with contributions from many others. They do cover similar ground however, especially when Al-Khalili meets mathematician Ian Stewart to examine one of the early texts on al-jabr; that is, algebra.

The conclusion of this episode is that by gathering texts from many different places, Islamic scientists proved that science is a universal concept that belongs to no one religion or culture; rather, it can be appreciated by everyone. No arguments here. I will say that at an hour, the programme was perhaps overly long. I can lay the same criticism against it as I did to The Story of Maths – less of our narrator wandering through generic marketplaces please! At least there was no dodgy CGI, however.

As I said at the start, I watched the programme on iPlayer, so of course so can you. If you liked The Story Of Maths, or perhaps if you missed it but want to learn about the history of science, I suggest you give it a look.

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University of Warwick mathematician Ian Stewart has provided New Scientist with a scientific guide to gift wrapping. Very festive. Professor Stewart informs us about the “sausage conjecture”, which asks what the most efficient way to wrap a group of circles or spheres is.

The tastiest way to wrap a sausage

For two and three dimensions, we have the answer: round circular objects (like mince pies) should be stacked end to end like a sausage if you have six or fewer, but for seven or more you’re better off arranging the pies in a hexagon and wrapping them that way in order to minimise the paper used. For spherical objects (Christmas puddings, of course) the split comes at 56 or fewer versus 57 or more.

So far, so simple, and good enough for anyone looking to wrap presents this Christmas. You might think we could just leave it there, but mathematicians never can. Extend the problem to four dimensions, and matters become predictably more complex. Now, you might be asking what a four-dimensional sphere looks like, and the truth is it’s impossible for the human mind to visualise. Mathematicians have no trouble with higher dimensions however – just add another number to your coordinate system. So, whilst we need two numbers to describe any point on a circle, and three numbers for a sphere, a group of four numbers will let us mathematically explore a so-called hypersphere.

How exactly do you go about wrapping a group of hyperspheres them? Well, for 50,000 or fewer you’re looking at a hyper-sausage, and for 100,000 you’re looking at something distinctly un-sausage-like – thought no-one knows exactly what. As for the specific trade off point, it isn’t as clear cut as with circles or spheres, but it definitely lies between 50,000 and 100,000 hyperspheres.

So what about the “sausage conjecture”? Unfortunately, it’s nothing to do with the trimmings at Christmas dinner, but rather states that for objects with five-dimensions or more, sausages are always best. This rather uninituive result, given the rules for two, three and four dimensions, was put forward in 1975 by Hungarian mathematician László Fejes Tóth.

Whilst it’s no Fermat’s Last Theorem or Riemann Hypothesis, some headway has been made with the sausage conjecture. In 1998 Ulrich Betke, Martin Henk and Jörg Wills proved that it was true for 42 or more dimensions, just leaving the cases 5 to 41. Perhaps you’d like to contemplate them as you wrap your Christmas presents!

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I’m sorry to have two similar stories so close together, but when I saw that the Sun had published a formula for determining if your boobline is too low, I just had to say something.

Apparently, following the dress-popping antics of Britney Spears “scientists, undies experts and mathematicians have been trying to figure out where the decency perimeter lies.” I’ll quote the “result” in full.

The equation is O=NP(20C+B)/75.

To figure out the naughtiness rating (O), you times the number of nipples exposed, from zero to two or expressed as fractions of nipple shown (N) with the percentage of exposed frontal surface area (P).

The sum in brackets is 20 multiplied by the cup size (C), where A cup is one, B is two, C is three and D or above is five.

Add that figure to B, the bust measurement in inches. Then divide your answer by 75. Any score higher than 100 is counted as obscene.

Can anyone spot the immediate problem with the equation? It’s this: if N is zero, then O will be zero, because anything multiplied by zero is zero. In other words, if no nipples are shown then the “naughtiness rating” will always be zero! Hardly scandalising, I think you’ll agree.

What’s worse is the Sun actually demonstrate this in the article, with their example calculation for Britney:

Britney’s tight fitting Roberto Cavalli dress showed off around 70 per cent of her breasts, and experts at Wonderbra think she is a 32D. Without any nipple exposure, Britney’s formula works out as 0x70x(20×5+32)/75 = 123.2.

They’ve clearly multiplied by zero, and yet got a non-zero number! What’s worse, the sub-editor who wrote the headline has substituted the O in the equation for a 0, rendering it completely meaningless. It’s a shame actually, because for once everything in this formula is quantifiable in an non-subjective manner. Don’t get me wrong, it’s a load of rubbish (why multiply the cup size by 20? Why is a score of 100 obscene), but I have to give whoever came up with this formula some small amount of credit for dealing in actual measurements.

That’s the other problem actually – who did come up with this? The Sun quote William Hartson, “who holds an MA in Maths from Cambridge University”, and is also the author of “Drunken Goldfish and Other Irrelevant Scientific Research”. Ah, I thought to myself – another book to shill – but no, Drunken Goldfish was published in 1987! I think the Sun may have just gone to Mr Hartson for an “expert” quote. A listing on another book at Amazon indicates that he writes “surreal humour” for the Daily Express. Further on in the article, a spokesperson from Wonderbra is quoted. Maybe they came up with the formula? It’s possible, but I can’t find any information indicating this to be the case.

Really, I’m over-thinking this. The article is little more than an excuse to publish pictures of scantily clad women, under the pretence of evaluating them with the formula. Sex sells papers, as is well documented on Just A Theory with what I like to call the Scarlett Johansson School of Science Reporting. Still, as you should’ve realised by now, I can’t resist a “formula for” story. Thankfully however, my reasons are the exact opposite of the mass media!

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Long term readers of Just A Theory may remember that one of the very first posts here was about a pet hate of mine: junk equations. Back then it was a formula for fame, but this time it’s the bane of students with essay deadlines ever: procrastination. Thankfully I handed in my essay yesterday, so I have some free time to rip in to this nonsense.

Professor Piers Steel has, according to the Telegraph spent “more than 10 years” studying why people procrastinate. Depending on who you ask, he’s either a psychologist or a business professor at the University of Calgary (the Telegraph say the former, the Daily Mail and the Times the latter).

On to the equation itself. It’s U = EV/ID, where U stands for “utlity”, or your desire to complete a given task. E is the expectation of succeeding in your task, whilst V is the value of completing it. I is the immediacy of the task, and finally D is your personal sensitivity to delay.

Well, that’s what the Telegraph says. The Daily Mail give a different formula: U = EVTC, where T is your tendency to delay work, and C the consequence of not completing it. By simple substitution, it must be that 1/ID = TC. Now, I can see an argument for saying that T has just been re-written as 1/D (in the same way that you can write 0.5 as 1/2), as they are both about delay, but how does the immediacy of the task (I) relate to the consequence of not completing it (C)? Already I’m starting to see the cracks in this equation…

For the definitive answer I went to Prof. Steel’s website, which provided me with the following:

Yet more variables! We’ve already met U, E, V and D, but now we have G (which seems to be standing in for the Greek letter Gamma which was actually used in the equation). Confusingly, G appears to be taking the place of D in the equation described by the Telegraph, whilst D here is now I. To avoid any further confusion, I will refer to Steel’s form of the equation, U = EV/GD from now on. To reiterate: E is expectancy of successful completion, V is the value of completion, G is the sensitivity to delay, and D is the immediacy of the task.

Besides changing variables like they were underpants, the problem with all of these formulas is that the values in them are completely unscientific and not at all measurable. Granted, your expectation of completing a task successfully could be expressed as a probability, for example, but such a measure is very subjective. What are the odds of getting an A for an essay? They simply can’t be calculated.

The other issue is the mathematical validity of the formula. If your sensitivity to delay is very low (and thus you have a small G), your utility value will be high – but surely it should be the other way around? If you don’t like to put things off, you’re less inclined to procrastinate! So maybe G should be measured from 1 to 10, with 1 being a high sensitivity and 10 being low. All this really illustrates is that it is very easy to come up with a formula for anything – as long as you fiddle the numbers to give that answer that you want!

Actually, it appears that this formula has more than one thing in common with the fame formula from my early post. Like that example, this equation is being used by its creator to publicise an upcoming book. Of course, all of the newspapers that have picked up this story are giving him a nice little bump of free advertising.

It shouldn’t need saying again, but I’m going to any way: these formula stories are a complete waste of time. They’re the absolute dregs of scientific journalism, and you shouldn’t pay any attention to them whatsoever. So, stop reading this and get back to work!

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Football statistics. The lengths of roads in Britain. Fundamental physical constants. What do all these groups of numbers have in common? The answer may surprise you, but it is this: in all three data sets, numbers that begin with 1 are far more common than those whose first digits are 2, 3 and so on up to 9.

Now, you might expect all numbers to start with 1 to 9 equally. As there are nine numbers you would think that chances of any number beginning with 1 would be 1/9, or around 11%, but actually it is more like 30%! So what makes these groups so special?

It’s a bit of a trick question actually. It turns out that for many large sets of data, numbers that start with a 1 crop up around 30% of the time. Numbers that start with a 2 occur around 18% of the time, and the probability decreases with each successive number. It’s not just in the examples I listed above that this happens, but also stock exchange data, population figures, and many more. This seemingly strange phenomenon is all thanks to Benford’s law.

It was first observed by a man named Simon Newcomb in 1881. As both a mathematician and an astronomer, he often used logarithms in his work. The logarithm of a number in a particular base is the power to which that base must be raised in order to produce that number. It’s easy to illustrate with an example: in base 10, the logarithm of 1000 is 3, because 103 = 1000. We write that log10(1000) = 3.

Logarithms can be used to make complex calculations much simpler, as long as you know how to convert back and forth to regular numbers. Nowadays we can let a computer do all the work, but back in Newcomb’s time people were forced to rely on weighty tombs of pre-calculated logarithms. One day, Newcomb realised that the pages of the book he was using became more worn the closer you were to the front. He came up with a formula that described the probabilities, as shown by this handy graph:

Benford's law in action: lower leading digits are far more common

Why then do we call this Benford’s law, if Newcomb came up with the formula? Well, Newcomb dismissed the idea an ideal curiosity. It was forgotten until 1938, when physicist Frank Benford noticed the exact same occurrence. He decided to investigate, and gathered masses of data to see if the rule was universal. Looking at sets similar to the ones described at the start, he found that it was really true: when it comes to large amounts of data, not all numbers are created equal.

Of course, you have to be careful when applying Benford’s law. A list of secondary school pupils and their ages will not follow the law, since all pupils must be aged 11-18 the probability of a leading 1 is 100%! On the other end of a scale, a collection of dice rolls will show that each number has a 1 in 6 chance of appearing; dies are truly random. Benford’s law will only apply in cases that fall somewhere between these two extremes, but thankfully this still includes a lot of data.

An interesting fact about Benford’s law is that it applies no matter the units of measurement used. You can measure your roads in miles or kilometres, and Benford’s law will still apply. This is known as scale invariance, and can actually be used to mathematically derive Benford’s law. You can work out which distributions of first digit probability stay the same when you switch from miles to kilometres, and it turns out there is only one: the formula that Newcomb came up with.

Benford’s law is a fascinating mathematical fact, but surprisingly it also has practical applications. Get this: it solves crime. No, really. If you’re a crooked accountant who likes to cook the books, Benford’s law will catch you out if you aren’t careful. If our dodgy dealer doesn’t know about Benford’s, they will probably pick numbers at random, or tend to stay in the “middle” (4, 5, 6, 7). Either approach will result in a data set that looks perfectly reasonable at a casual glance, but an analysis with Benford’s law will reveal that not enough numbers start with 1.

There you have it. Benford’s law: a mathematical oddity that just happens to have its uses. For fun, I thought I’d see how Just A Theory page views stack up against Benford’s law. WordPress can track how many times each page has been visited, so I grabbed the first digit from each of these numbers. The results:

It’s not perfect, but Just A Theory doesn’t have that many pages (yet!). As with anything statistical, a larger data set will get you closer to a mathematically predicted distribution. Still, it’s pretty impressive to see Benford’s law in action. Maybe I’ll try again in a year’s time, when I have more data!

Bletchley Park, home to the Allied codebreakers of World War II, has secured a grant of £330,000 to restore the roof of the historic site. The Grade II-listed mansion is at risk due to previous neglect.

Codebreakers who were at Bletchley include Alan Turing, arguably the founder of computer science. The need to crack the German Enigma machine lead to great developments in cryptoanalysis and other sciences. It’s a fascinating place that I’d love to visit one day, so hopefully this new money will help preserve the site.

The news follows on from China’s previous space efforts at the end of September, in which they broadcast footage of a first space-walk back to those watching on Earth. It could also be seen as an answer to the American’s testing their latest moon buggy prototype.

China says that its lunar mission will include three steps of “orbiting, landing and returning”, but has not yet set any dates for manned moon mission yet.

Not lead into gold, but tequila into diamonds

Mexican scientists have discovered a way to turn tequila into diamonds. It turns out that the chemical makeup of the drink has a ratio of hydrogen, oxygen, and carbon atoms which places it within the “diamond growth region.”

The scientists turned to tequila not for its intoxicating quality, but because previous efforts to create diamonds from organic solutions such as acetone, ethanol, and methanol had proved unsuccessful. They then realised that their ideal compound of 40% ethanol and 60% water was remarkably close to tequila.

Luis Miguel Apátiga was one of the researches from the National Autonomous University of Mexico:

“To dissipate any doubts, one morning on the way to the lab I bought a pocket-size bottle of cheap white tequila and we did some tests,” Apátiga said. “We were in doubt over whether the great amount of chemicals present in tequila, other than water and ethanol, would contaminate or obstruct the process, it turned out to be not so. The results were amazing, same as with the ethanol and water compound, we obtained almost spherical shaped diamonds of nanometric size. There is no doubt; tequila has the exact proportion of carbon, hydrogen and oxygen atoms necessary to form diamonds.”

The diamonds were made by heating tequila to transform it into a gas, and then heating this gas further to break down the molecular structure. The result: solid diamond crystals, about 100-400 nanometres in size. They could be used to coat cutting tools, or as high-power semiconductors, radiation detectors and optical-electronic devices.

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Well, not quite, but close. Notices of the American Mathematical Society have published details of computer programs that can provide rock-solid mathematical proofs.

This is extremely important, because in maths, proof is king. You could count prime numbers (which can only be divided by one or themselves) until the proverbial cows come home, and by the time you get to one squillion – not actually a real number, but let’s say it’s pretty big – you might be satisfied to say there were an infinite number of primes. Not so the mathematician, who will only be convinced by a logical proof.

The trouble is, even in the most basic proof you have to make some assumptions of previous results in the field. It doesn’t really matter because a sufficiently advanced reader will be able skip over these leaps of logic, but some theorems become some long and complex that even without dotting all the mathematical “i”s the proof can reach hundreds of pages long.

Checking such a proof would be incredibly arduous, but for the mathematician it must be done. This is where computers come in. A computer can develop a “formal proof”, in which every single statement is checked all the way back to first principles.

We’re not even talking 1 + 1 = 2 here. The Principia Mathematica, a seminal work on the foundations of mathematics published nearly a century ago, does not reach a proof of 1 + 1 = 2 until page 379. And mathematicians use pretty small fonts.

This demonstrates how ridiculous it would be to create such a formal proof by hand. It would be like providing a full dictionary definition of each word in this blog post along with the text – madness. Yet, for mathematicians to be truly, truly sure, a formal proof is the only way to go.

The computers aren’t quite there yet. Mathematicians still have to break the proofs down before they are fed into the program, so there we’re not quite at the level where your PC can leaf through the latest mathematical journals. It is possible, however, to let the computers “explore” the mathematics on their own – and perhaps even come up with some points the humans may have missed.

Ultimately, mathematicians would like to have formal proofs of at least the most important theorems. Thomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin to “the sequencing of the mathematical genome”. Impressive stuff.

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Last night BBC 4 broadcast the first episode of a new four part series entitled The Story of Maths. It’s presented by Marcus du Sautoy, Oxford professor and pop-sci mathematician extraordinaire, who takes a look at the history of maths and why it is so important. This initial outing focuses on the three ancient civilizations who were the founders of maths: the Egyptians, the Babylonians, and the Greeks.

The Egyptians were practical problem solvers, and their need for bureaucracy and land management lead to the development of a counting system. Common problems, such as how to split nine loaves of bread between 10 people, were worked out in detail, but the Egyptians never realised the power of a generalised proof, forcing them instead to work out the same problem multiple times, but with different numbers. As he walks around a modern Egyptian market, and marvels at the Pyramids, du Sautoy demonstrates some of their ancient methods. (For those still wondering, each person receives one half, one third, and one fifteenth of a loaf.)

The Babylonians used maths to solve every day problems as well, but they also taught more generalised solutions in schools. Most of the mathematical records we have from those times are actually preserved clay tablets that record the workings of school children. They knew about quadratic equations like x2 + 3x + 2 = 0, and du Sautoy blames the “recipes” used to solve such problems for poor maths teaching in modern classrooms.

Finally, we get to the Greeks, who in du Sautoy’s opinion are the true founders of maths – they were the inventors of proof, which opened up “a gulf between the other sciences” and are as true today as they were 2,000 years ago (a point he feels the need to make twice).

It’s a good primer to early maths, and I imagine it will be the most accessible programme of the series, since mathematics is a field that builds on its past and becomes increasingly complex. As one of the talking heads points out, Greek mathematics is still taught in schools today – because more modern concepts are completely inaccessible. Even at undergraduate level I spent most of my time learning about the 17th and 18th centuries; the 1970s were about the upper limit. This does make me wonder whether the series will remain engaging to the average viewer as it reaches more modern times.

I only have one criticism and it’s nothing to do with du Sautoy, who was excellent as always. It might be a small quibble, but the computer graphics used to illustrate his narrations were absolutely terrible. As du Sautoy was sent flying around on slices of Pyramid and hot air balloons, I found it increasingly difficult to concentrate on what he was saying, as all I could think about was how cheap and cheesy looking the animations were. Seriously, they would not have looked out of place a decade ago. It seems silly to knock the programme for this reason, but production values are an important part of getting your message across, and doing it badly just doesn’t help.

Next week, du Sautoy heads east. I expect we will be hearing about Chinese and Arabic mathematicians, along with algorithms and the number zero. It should be interesting, and I do recommend you watch this first episode, despite the dodgy CGI.

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Playing a video game for 20 minutes a day can increase your mathematical potential, a study by Learning and Teaching Scotland has found. Apparently a daily dose of Brain Training on the Nintendo DS helped Scottish school children gain higher scores on their maths tests.

For the uninitiated, Brain Training is a fairly simple game that challenges players with short tests such as mental arithmetic or counting. The idea is to play the game daily with a view to improving your “Brain Age”, a fairly unscientific measure of how “young” your brain is. It’s pretty popular – even Nicole Kidman is at it – but can it really improve your thinking power?

To find out, over 600 pupils in 32 primary schools were given a maths test at the beginning of the study. For the next nine weeks, those in the control group received their normal teaching, whilst the other group were given 20 minutes of Brain Training at the start of each day. At the end of the study period the pupils were tested again, and the two scores compared. The control group showed some improvement, but those training their brains saw a further increase of 50%, from an average of 78 to 83 out of 100. They were also able to solve problems faster, dropping five minutes from an average 18.5 to 13.5 off their total test time.

Interestingly, children who were less competent at maths found the game more beneficial than their more able classmates, showing a larger increase in test scores overall. It could be that they find this non-traditional method of teaching more engaging than their standard lessons. The research also showed that both girls and boys benefited equally from using the game.

All positive results then, but will we be seeing Brain Training in classrooms an time soon? Unfortunately, I think the cost of equipment might prove to be prohibitive. The researchers who carried out the study make it clear they did not receive any financial aid from Nintendo, so presumably they forked out for the game and console themselves. Brain Training sells for around £15, whilst a Nintendo DS is close to £100. For a typical primary class of about 28 pupils, that works out at about £3200.

It makes me wonder if this would be the most cost-effective method of improving pupils mathematical ability, and perhaps more research is needed to find the teaching method with best “pound per percentage-point” ratio. Still, if you’d like to have fun and improve your mind at the same time, it could be that Brain Training is just the game for you. Personally, I think I’ll stick to Super Mario.

The type of music you like could be linked to your personality, suggests a study carried out by Professor Adrian North of Heriot-Watt University. Apparently fans of country and western are “hardworking, outgoing” whilst indie lovers are “low self-esteem, creative, not hard working, not gentle”. Sounds like a bunch of nonsense to me – what if you like both country and indie? I haven’t been able to find a published paper on the research, which might validate it a little more, but I’m not holding my breath.

Because I say so

In the latest of a series on statistics in the media, Michael Blastland talks about the pitfalls of causation and correlation. Just because event A occurred before event B, it does not mean that A caused B – and yet so many stories in the media report just that. One you should always watch out for, so have a read.
Fruit for thought

Finally, some amazing photos of fruit taken using a scanning electron microscope. The colours may be false, but its all still very pretty.

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You hopefully now understand the concepts that make up Euler’s equation, so let’s move on to how the equation arises. You may have already realised that for eiπ + 1 = 0 to be true, it must also follow that eiπ = -1, simply by rearranging the equation.

Think about that for a second. You’ve got a combination of three extraordinary numbers – e, i and π – numbers that seem to have no relation what so ever, and they combine to make -1. How can this be? Why don’t these numbers create a horrible decimal like 8.23487 or similar? Two of the numbers, e and π, cannot even be written down in full because they are infinity long, and yet shove i into the mix and we get the simple, beautiful result: -1.

It all stems from the exponential function, ex. A function is basically a rule for turning one number into another. You take your independent variable, x, and plug it into your function to get the dependant variable. A simple function might be f, where f(x) = x2. If you stick x = 2 into f, out pops f(2) = 22 = 4. The exponential function works the same way, just replace x with whatever number you are interested in. For Euler’s equation, x = iπ gives us the desired result.

Some functions can be expressed as lots of other functions – an infinite number of functions, in fact. This representation is often of a type known as a Taylor series, and for ex it looks like this (I’ve borrowed a graphic from Wikipedia to make it clearer):

I’ve mentioned factorials like 2! and 3!before, but a quick reminder: n! just means “multiply together all the numbers from 1 to n. The ellipsis at the end of the equation means that the pattern goes on forever. “Well how does that help?” you ask. “Now I’ve got an infinite amount of things to deal with, and we don’t seem to be getting any closer to -1!” Fear not.

You see, ex is actually hiding two other functions you may remember from school – the trigonometric duo, sinx and cosx. These two help you work out the length of a triangle’s sides from its angles, and they crop up everywhere in mathematics. When you stick x = iy (y is just another variable, like x) into our exponential function, it turns out that the Taylor series above becomes equal to two series added together – the series for cosy and isiny.

This means that eiy = cosy + isiny. Like so many things in mathematics, this equation has an alternate, geometric representation – in this case, eiy traces out a circle in the complex plane, a way of representing both real and imaginary numbers. Once more, Wikipedia has an excellent representation. They have used the Greek letter phi rather than the y I use here, but it means the same.

We’re nearly there now. We just need to set y = π, which represents a turn half way around the circle. This also happens to place us on the point where the circle intersects the real numbers at -1; in other words, eiπ = -1. This is because cosπ = -1 and isinπ = 0. This what Euler realised, and although it is thought that he never actually wrote the expression eiπ + 1 = 0, it follows directly from his result.

Euler’s equation summarises addition, multiplication, and exponentiation using five of the most important numbers in mathematics; e, i, π, 1, 0. It’s a truth universal in any language and to any culture. There is quite journey to get the equation, but one I believe is well worth travelling. I hope my explanation here leads you to agree.

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In the very first post on Just a Theory I mentioned Euler’s equation, considered by many to be the most beautiful equation in the whole of mathematics. I decided to share with you everything I know about this wonderful equation, eiπ + 1 = 0. Let’s start with some of the more unfamiliar elements.

π: You’ve probably met π, written Pi and pronounced “pie”, before and perhaps you remember that it is the number you get when dividing the circumference of a circle by its diameter. This neat little Wikipedia animation demonstrates the principle. Pi has an infinite number of digits because it is an irrational number; this means that it cannot be represented by a simple fraction such as 1/2 or 365/789. I’m hopeless at remembering digits of Pi and normally stick to 3.14.

i: What is the square root of 4? In other words, what number must you multiply by itself to get 4? The answer is of course 2. What about the square root of -4? It can’t be -2, because -2 is also the square root of 4 – since multiplying two negative numbers results in a positive number. The answer is 2i, because i is defined as the square root of -1. i is an imaginary number – but that doesn’t mean it’s just made up. For example, many problem in electrical engineering can only be expressed using i.

e: Another irrational number like π, e is the base of the natural logarithm. What that means isn’t really important in this context – just think of it as another important mathematical constant like π. The numerical value of e is approximately 2.718.

Exponentiation: You should be familiar with the concept of raising one number to the power of another, such as 23 = 2 x 2 x 2 = 8. This is exponentiation. What does the strange beast eiπ represent then? If you are trying to multiply e by itself iπ times you might be left scratching your head, but all will become clear as Euler’s equation is explained.

What with it being a bank holiday, I’ll give you a break and leave it there for today. Hopefully you’ve understood these building blocks, and your ready to tackle the full equation tomorrow. If not, leave a message in the comments and let me know if you need further clarification.

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If I remember anything from my days of learning foreign languages, it’s how to count. Not very impressive I grant you, but I can still knock out an “un, duex, trois” or an “ein, zwei, drei” when required. Counting is such a basic and universal skill that it is hard to imagine life without it, but certain aboriginal communities do not have words or gestures to represent numbers. A study by University College London and the University of Melbourne of children from two such communities has found the lack of words is not a hindrance to counting.

The study looked at children aged four to seven from two aboriginal groups, one speaking a langage called Warlpiri whilst the other used Anindilyakwa. Both have words for one, two, few and many, and Anindilyakwa uses numbers up to 20 in rituals but children are not taught these. As a control group the team also worked with an English-speaking indigenous community.

Professor Brian Butterworth of the UCL Institute of Cognitive Neuroscience was lead author of the study, and details the difficulty in designing questions that the children could answer:

“In our tasks we couldn’t, for example, ask questions such as “How many?” or “Do these two sets have the same number of objects?” We therefore had to develop special tasks. For example, children were asked to put out counters that matched the number of sounds made by banging two sticks together. Thus, the children had to mentally link numerosities in two different modalities, sounds and actions, which meant they could not rely on visual or auditory patterns alone. They had to use an abstract representation of, for example, the fiveness of the bangs and the fiveness of the counters. We found that Warlpiri and Anindilyakwa children performed as well as or better than the English-speaking children on a range of tasks, and on numerosities up to nine, even though they lacked number words.

It appears being able to count is an innate skill. This could explain why children with dyscalculia, a form of dyslexia relating to mathematics, find arithmetic so difficult to learn. Even with our counting system of “one, two, three” to aid them, a lack of this innate skill causes sufferers to struggle. Professor Butterworth is conducting another study in order to find the differences in brains of people with the disorder.

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Andrew Hodges’ inspiration for the title One to Nine was Sudoku, the immensely popular number puzzle. Hodges comments that newspapers insist the puzzles require no mathematical knowledge, in order to not scare away an often maths-phobic British public – indeed, Sudoku does not even require numbers, since substituting nine letters or symbols into a puzzle would leave the logic required to solve it unchanged.

Hodges describes logic as one of the most fascinating elements of ‘adult mathematics’, wholly different to the ‘school maths’ that newspapers try to distance themselves from. The book aims to provide an insight into this for those who may have been turned off the subject at school.

Unsurprisingly, the book is split into nine chapters, One through Nine. Each begins with a characterisation of the number; seven ‘needs sifting and sorting out’, whereas three ‘doesn’t just talk’, but ‘thinks big’. The chapter titles are a bit of a gimmick at times. Six is the first perfect number, so-called because 6 = 1 + 2 + 3 = 1 x 2 x 3, and this leads to a discussion of factorials. Six is 3! (pronounced ‘three factorial’) because 3! = 1 x 2 x 3., and the factorial of a number n is simply the product of all numbers from 1 to n. The chapter continues with probabilities, the Enigma machine, and Euler’s equation – all very interesting topics with links to factorials, but do they really relate particularly to six, more so than any other number?

Gimmicks aside, One to Nine is a whistle-stop tour of pop-sci mathematics, with sections ranging from black holes to game theory to musical harmony. Each topic is well described and often accompanied by many useful diagrams, although some appear to have been lifted straight from a .jpg file, complete with ugly compression artefacts – a bit more care could have been take in order to provide high quality images.

Numerous equations may discourage the casual reader, but they are always accompanied by a thorough explanation in the text. Stephen Hawking was told when writing A Brief History of Time that ‘each equation in the book would halve the sales’; I hope this is not the case else I will have already lost 75% of my readership! For those who really will not abide equations, relax – they can for the most part be skipped.

Sprinkled throughout the text are problems rated on a Sudoku-like scale, from GENTLE to DEADLY. I found these to be a welcome addition, but normally skipped over any that I was unable to solve in a minute or two, so as not to slow down the pace of the book. Placing these at the end of each chapter would have made me more inclined to give them a go.

Helpfully, all of the solutions are provided on the website for the book, along with further notes and comments. Unfortunately the book does not feature a bibliography or recommended reading list, so if you do become engrossed in a particular topic you will have to hunt out more information by yourself, but the website does go some way to assisting with this.

If you would like to learn how mathematics is used in a variety of scientific fields and are not too afraid of a few equations, One to Nine is a good place to start. In fact, Hodges’ appropriation of Sudoku is quite apt. If you enjoy the use of logic in a Sudoku puzzle, but have dreaded school memories of multiplication tables, perhaps One to Nine can show you the world through the filter of ‘adult mathematics’.

The latest culprit is PR agent Mark Borkowski who claims to have found a “scientific formula” for fame. The formula itself is given as follows:

F(T) = B+P(1/10T+1/2T2)

where:

F is the level of fame;

T is time, measured in three-monthly intervals. So T=1 is after three months, T=2 is after six months, etc. Fame is at its peak when T=0. (Putting T=0 into the equation gives an infinite fame peak, not mathematically accurate, perhaps, but the concept of the level of fame being off the radar is apposite.);

B is a base level of fame that we identified and quantified by analysing the average level of fame in the year before peak. For George Clooney, B would be a large number, but for a fabulous nobody, like a new Big Brother contestant, B is zero;

P is the increment of fame above the base level, that establishes the individual firmly at the front of public consciousness.

Not that it really matters, but this is terribly unclear. A more correct way to write it would be F(T) = B+(1/(10T)+1/(2T^2))*P, eliminating any ambiguity as to what each symbol means, but as with all of these stories scientific accuracy is not high on the agenda. Borkowski has made the same two mistakes that always crop up in these formulas – unmeasurable variables and confirmation bias.

The unmeasurable variables in this case are F, B, and P. T is time, where the units of T are periods of 3 months – not exactly orthodox, but still completely measurable. F, B and P however are measures of fame, for which I know of no scientific units. Perhaps fame is measured in the units of star power – solar luminosity.

Yes, I’m being facetious, but it is an important point. One of the greatest tools available to a scientist are the standard units of measurement known as SI units. I’ll talk about them in more depth another day, but they include metres, kilograms, and seconds – quantities we are all familiar with. This common set of units allow scientists to communicate their findings in a meaningful way, and the results of not confirming your units can be disastrous, as NASA discovered when they mixed up feet and metres, causing an unmanned spaceship to crash.

The other problem, confirmation bias, is an interesting one. It basically amounts to “people believe what they want to believe”, and it’s definitely in action here. Borkowski wanted to match Andy Warhol’s 15 minutes of fame with his own 15 months of fame:

I started to wonder if Andy Warhol – an artist by calling but a master of the stunt and the soundbite – was right; does everyone get 15 minutes of fame? It occurred to me that it should be possible to look at fame statistically, to analyse the evidence we have all witnessed in the media, to see if fame’s decline can be quantified. The answer, I discovered, is that it can be, and that Warhol was partially right – but the first spike of fame will last 15 months, not 15 minutes.

In looking at fame “statistically”, it turns out that 15 months is exactly right! Well done, Borkowski.

This formula fits the data remarkably well, giving a precise numerical value to the 15-month theory: if I put in T=5 (corresponding to 15 months after the peak), it gives F=B+P(1/50+1/50), which works out at F=B+.04P. In other words, up to 96% of the fame-boost achieved at the peak of public attention has been frittered away, and the client or product is almost back to base level.

Of course, if you put in T = 6 (i.e. 18 months) you get F = B + 0.03P (rounding off the decimal point). Three months later, it appears our Big Brother contestant hasn’t really got much less famous than they were after 15 months. What about after two years, when T = 8? In that case, F = B + 0.02P – fame doesn’t really appear to be dropping off very quickly, does it? The claim that ‘the study showed pretty conclusively that any specific boost to fame is sustained for approximately 15 months…’ isn’t remotely conclusive – in fact, I’ve just show that you can reach an entirely different conclusion by choosing different values of T.

The reason I hate these formula stories with a passion is that they damage the public perception of science. The ideas they offer are meaningless, suggesting that all scientist do is sit around dunking biscuits in a quest for perfection. That story is nearly a decade old, so the junk equation is clearly not a new concept, and I don’t think we will be rid of it any time soon. So, the next time you see an article proclaiming that science has once more advanced, and we now know how to calculate the cuteness of puppies or the magic of rainbows, please do the only sensible thing – ignore it.